where \(tr\)  is the usual trace of a Hilbert space operator and \(I_W\) is the standard projection \(l^2(V) \twoheadrightarrow l^2(W)\) .
Moreover, we may commute the operators inside the trace: \(tr(E(B)I_W) = tr(I_W E(B))\)  (see [1]}).
From the spectral theorem, it follows that \(dk(\cdot )\)  is a measure on \(sp(\Delta )\) .
It is well known (for instance see [2]}) that this measure is purely continuous except a set of point masses which occur precisely at the eigenvalues of \(\Delta \) .
When \(\lambda \)  is an eigenvalue, \(dk(\lbrace \lambda \rbrace )\)  is called the von Neumann dimension of the eigenspace \(E_\lambda \)  (see [2]} for further context)
and when \(\Gamma = \Gamma (G,S)\)  is a Cayley graph, \(dk(B) = <E(B) \delta _1, \delta _1>\)  and is often called the spectral measure of \((G,S)\) .
